does risk seeking drive asset prices - post and levy 2005

7/31/2019 Does Risk Seeking Drive Asset Prices - Post and Levy 2005

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DOES RISK SEEKING DRIVE ASSET PRICES?

A STOCHASTIC DOMINANCE ANALYSIS OF AGGREGATE INVESTOR

PREFERENCES

THIERRY POST AND HAIM LEVY

ERIM REPORT SERIES RESEARCH INMANAGEMENT

ERIM Report Series reference number ERS-2002-50-F&A

Publication May 2002

Number of pages 36

Email address corresponding author [email protected]

Address Erasmus Research Institute of Management (ERIM)

Rotterdam School of Management / Faculteit Bedrijfskunde

Rotterdam School of Economics

Erasmus Universiteit Rotterdam

P.O. Box 1738

3000 DR Rotterdam, The Netherlands

Phone: + 31 10 408 1182

Fax: + 31 10 408 9640

Email: [email protected]

Internet: www.erim.eur.nl

Bibliographic data and classifications of all the ERIM reports are also available on the ERIM website:www.erim.eur.nl
http://www.erim.eur.nl/http://www.erim.eur.nl/


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ERASMUS RESEARCH INSTITUTE OF MANAGEMENT

REPORT SERIES

RESEARCH IN MANAGEMENT

BIBLIOGRAPHIC DATA AND CLASSIFICATIONS

Abstract We investigate whether risk seeking or non-concave utility functions can help to explain the

cross-sectional pattern of stock returns. For this purpose, we analyze the stochastic dominance

efficiency classification of the value-weighted market portfolio relative to benchmark portfolios

based on market capitalization, book-to-market equity ratio and momentum. We use various

existing and novel stochastic dominance criteria that account for the possibility that investors

exhibit local risk seeking behavior. Our results suggest that Markowitz type utility functions, withrisk aversion for losses and risk seeking for gains, can capture the cross-sectional pattern of

stock returns. The low average yield on big caps, growth stocks and past losers may reflect

investors twin desire for downside protection in bear markets and upside potential in bull

markets.

5001-6182 Business

4001-4280.7 Finance Management, Business Finance, Corporation Finance

Library of Congress

Classification

(LCC) HG 4529.5 Portfolio management

M Business Administration and Business Economics

G 3 Corporate Finance and Governance

Journal of Economic

Literature

(JEL) G 12 Asset Pricing

85 A Business General

220 A Financial Management

European Business Schools

Library Group

(EBSLG) 220 P Investments, portfolio management

Gemeenschappelijke Onderwerpsontsluiting (GOO)

85.00 Bedrijfskunde, Organisatiekunde: algemeen

85.30 Financieel management, financiering

Classification GOO

85.33 Beleggingsleer

Bedrijfskunde / Bedrijfseconomie

Financieel management, bedrijfsfinanciering, besliskunde

Keywords GOO

Portfolio-analyse, Ris icoanalyse, Stochastische modellen

Free keywords Asset Pricing, Risk Seeking, Specification Error, Prospect Theory, Stochastic Dominance


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1

Does Risk Seeking Drive Asset Prices?

A Stochastic Dominance Analysis of Aggregate Investor

Preferences

THIERRY POST AND HAIM LEVY*

3RD VERSION, MAY 2002

1 Haim Levy is with the School of Business, The Hebrew University of Jerusalem, Israel. Post is

corresponding author: Erasmus University Rotterdam, P.O. Box 1738, 3000 DR, Rotterdam, The

Netherlands. We appreciate the comments by Roger Bowden, Winfried Hallerbach and Jaap Spronk, as

well as participants at a Center for Economic Studies seminar at the Catholic University of Leuven, the

30th EURO Working Group on Financial Modeling, and the 29 th Annual meeting of the European

Finance Association. Martijn van den Assem and Peter van der Spek provided invaluable research

assistance. We thank Mark Carhart for providing the data on the 27 Carhart benchmark portfolios. Postgratefully acknowledges financial support by Tinbergen Institute and Erasmus Research Institute of

Management.


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We investigate whether risk seeking or non-concave utility functions can help to explain

the cross-sectional pattern of stock returns. For this purpose, we analyze the stochastic

dominance efficiency classification of the value-weighted market portfolio relative to

benchmark portfolios based on market capitalization, book-to-market equity ratio and

momentum. We use various existing and novel stochastic dominance criteria that account

for the possibility that investors exhibit local risk seeking behavior. Our results suggest

that Markowitz type utility functions, with risk aversion for losses and risk seeking for

gains, can capture the cross-sectional pattern of stock returns. The low average yield on

big caps, growth stocks and past losers may reflect investors twin desire for downside

protection in bear markets and upside potential in bull markets.

THE TRADITIONAL MEAN-VARIANCE CAPITAL ASSET PRICING MODEL (MV CAPM) by

Sharpe (1964) and Lintner (1965) fares poorly in explaining observed cross-sectional

stock returns. Specifically, market beta seems to explain only a small portion of the

cross-sectional variation in average returns, while factors like market capitalization

(size), book-to-market equity ratio (BE/ME) and momentum systematically appear to

affect asset prices (see e.g. Fama and French, 1992, and Jagadeesh and Titman, 1993).

Related to this, the value-weighted market portfolio of risky assets seems highly

mean-variance inefficient, and it is possible to achieve a substantially higher mean

and a substantially lower variance with portfolios with a higher weight of small caps,

value stocks (high BE/ME stocks) and past winners; see e.g. Wang (1998).

One way to extend the MV CAPM is by changing the maintained assumptions

on investor preferences. If we do not restrict the shape of the return distribution, then

MV CAPM is consistent with expected utility theory only if utility takes a quadratic

form. (Less restrictive assumptions are obtained if we do restrict the shape of the

return distribution; see e.g. Berk (1997)). Extensions of the MV CAPM can beobtained by using alternative classes of utility. For example, Friend and Westerfield

(1980) and Harvey and Siddique (2000) assume that utility can be approximated using

a third-order polynomial, and Dittmar (2002) uses a fourth-order polynomial. 2 The

higher-order polynomials better fit stock return data than the standard quadratic utility

functions do. Still, the market portfolio remains inefficient and size, value and

2Similarly, Bansal and Viswanathan (1993) and Chapman (1997) use polynomial approximations inthe context of Arbitrage Pricing Theory and consumption-based CAPM respectively.


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momentum effects remain. Two potential problems of the extended models may help

to explain this result:

1. Risk seeking. The extended models typically maintain the assumption that

investors are globally risk averse and that utility is everywhere concave (i.e.

marginal utility is diminishing). However, there is evidence that decision-makers

are not globally risk averse, but rather they exhibit local risk seeking behavior

(i.e. the utility function has convex segments). For example, Friedman and

Savage (1948) and Markowitz (1952) argue that the willingness to purchase both

insurance and lottery tickets implies that marginal utility is inc reasing over a

range (see Hartley and Farrell, 2001, for a recent discussion). Similarly, active

stock traders seem to play negative-sum games and their behavior is sometimes

best described as gambling (see e.g. Statman, 2002). In addition, psychologists

led by Kahneman and Tversky (1979) find experimental evidence for local risk

seeking behavior.

2. Specification error. A difficulty in changing the preference assumptions is the

need to give a parametric specification of the functional form of the utility

function or to specify the maximum order of the approximating polynomial.

Unfortunately, economic theory gives minimal guidance for functional

specification, and there is a substantial risk of specification error. For example,

the fourth-order polynomial used in Dittmar (2002) implies that investors care

only about the first four central moments of the return distribution (mean,

variance, skewness and kurtosis). This approach is problematic if investors care

about the higher central moments or about lower partial moments (see e.g. Bawa

and Lindenberg, 1977), which generally cannot be expressed in terms of the first

four central moments. Another problem associated with low order polynomials is

the difficulty to impose restrictions on the derivatives that apply globally. For

example, we cannot impose nonsatiation by restricting a quadratic polynomial to

be monotone increasing and we cannot impose risk aversion by restricting a cubic

polynomial to be globally concave (see e.g. Levy, 1969).

To circumvent these problems, we may use criteria of Stochastic Dominance (SD; seee.g. Levy, 1992, 1998). Attractively, SD criteria do not require a parameterized utility


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function, but rather they rely only on general preference assumptions.3 Put differently,

SD effectively considers the full house of all moments of the return distribution rather

than a finite set. The SD literature involves a multitude of different criteria, associated

with different sets of preference assumptions. The traditional First-order Stochastic

Dominance (FSD) criterion assumes only non-satiation i.e. utility is monotone

increasing. The Second-order Stochastic Dominance (SSD) criterion adds the

assumption of global risk-aversion. Recently, a number of intermediate criteria have

been developed based on non-concave utility functions. Most notably, Levy (1998)

developed Prospect Stochastic Dominance (PSD), which assumes a S-shaped utility

function that is convex for losses and concave for gains. In addition, Levy and Levy

(2002) developed Markowitz Stochastic Dominance (MSD), which assumes a reverse

S-shaped utility function that is concave for losses and convex for gains.

In this paper, we use various existing and novel SD criteria to analyze asset

pricing. To focus on the role of preference assumptions, we largely adhere to the

remaining assumptions of the MV CAPM: we use a single-period, portfolio-oriented

model of a frictionless and competitive capital market with a large number of

expected utility investors. Within this model, we test whether the value-weighted

market portfolio is efficient relative to benchmark portfolios formed on size, BE/ME

and momentum. For this purpose, we assume a simple data generating process with a

serially independent and identical distribution for the excess returns. Of course, there

are good reasons to doubt our maintained assumptions, and to believe that our results

are affected by these assumptions in a non-trivial way. Still, we believe our approach

is useful, as we have to walk before we can run, and the analysis can form the

starting point for further research based on more general economic and statistical

assumptions.

The remainder of this paper is structured as follows. Section I introduces a

general SD efficiency criterion and it discusses the special cases used our study.

Section II discusses the issue of empirical testing. Specifically, we develop a general

3 For the sake of analytical simplicity, we phrase in terms of expected utility theory. However, SD rules

are economically meaningful also for many non-expected utility theories that account for e.g.

subjective probability distortion (see e.g. Starmer, 2002). For example, it is easily verified that the

MSD efficiency criterion is not affected by subjective transformations of the CDF that are increasing

and concave over losses and increasing and convex over gains, and hence MSD allows for subjective

overweighing of small probabilities of large gains and losses and underweighing of small and

intermediate probabilities of small and intermediate gains and losses. Interestingly, empirical studiessuggest that this reverse S-shape is the most common pattern of probability transformation (see e.g.

Tversky and Kahneman, 1992).


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Linear Programming (LP) test for fitting SD efficiency criteria to empirical data and

we derive the asymptotic sampling distribution of the test results. This section

effectively generalizes Posts (2001) treatment of SSD efficiency towards our general

SD efficiency criterion. Section III presents the empirical application. Finally, Section

IV gives concluding remarks and suggestions for further research. The Appendix

gives the formal proofs for our theorems.

I. STOCHASTIC DOMINANCE EFFICIENCY CRITERIA

We consider a single-period, portfolio-based model of a competitive capital market

that satisfies the following assumptions:

Assumption 1 The investment universe consists ofNassets, one of which is a riskless

asset. Throughout the text, we index the assets by {}Nii 1= . The excess returns

Nx are treated as random variables with a continuous joint cumulative

distribution function (CDF) : [0,1]NG .4 Investors may diversify between the

assets, and we will use N for a vector of portfolio weights. The portfolio

possibilities are represented by the simplex }1: = + eN .5

Assumption 2 Investors select investment portfolios tomaximize the expected

value of a once directionally differentiable, strictly increasing utility function

:u that is defined over portfolio return x .6 We represent all admitted

utility functions by { } xxuuU 1)(:0 , with

)()(lim)(

0

xuxuxu

+

for the directional derivative or marginal utility at x .7

4 Throughout the text, we will use N for anN-dimensional Euclidean space, and N+ denotes the positive

orthant. To distinguish between vectors and scalars, we use a bold font for vectors and a regular font for

scalars. Further, all vectors are column vectors and we use x for the transpose ofx . Finally, e is a unity

vector with dimensions conforming to the rules of matrix algebra.5 By using the simplex , we exclude short selling. Short selling typically is difficult to implement inpractice due to margin requirements and explicit or implicit restrictions on short selling for institutional

investors (see e.g. Sharpe, 1991, and Wang, 1998). Still, we may generalize our analysis to include

(bounded) short selling. In fact, the analysis applies for an arbitrary polytope if we replace with theset of extreme points of the polytope.6

We use a directional derivative to allow for kinked utility functions, including piecewise-linear utility

functions (see the proof to Theorem 1).

7 0U restricts marginal utility to exceed unity. This restriction effectively standardizes the test statistic

),( (see Section II) and forces it away from zero if the evaluated portfolio is inefficient. Still, the


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We may characterize different SD efficiency criteria by different classes of utility

functions, characterized by different sets of (conditional) linear restrictions on

marginal utility. Formally, we will denote different classes of utility functions by

{ }RryxyuxuUuU r ,,1),()()(:)( 0 L= . Here, { }R

rr 1= , with

2r for a polyhedron that represents the r-th restriction on marginal utility

(special cases are given below). Using this notation, we may define the following

general SD efficiency criterion:8

DEFINITION 1 Portfolio is )(U -SD efficient if and only if it is optimal

relative to some utility functions )(Uu , i.e.

(1) 0)()()()(maxmin)(

=

xxxx GuGu

Uu

.

Assumption 3 The value-weighted market portfolio of risky assets, say , is

)(U -SD efficient.

In asset pricing theories, efficiency of the market portfolio generally is not an

assumption but rather a prediction that follows from underlying assumptions on the

return distribution and investor preferences. For example, following Rubinstein

(1974), efficiency of the market follows from assuming that the preferences of the

different investors are sufficiently similar. In this case, we may use the utility function

of a representative agentwhose preferences are an aggregate of the preferences of theactual investors. In this paper, we do not take this route, because detailed distribution

and preference assumptions are not consistent with the SD approach of using minimal

assumptions. Rather, our analysis builds on revealed preferences. Specifically, our

restriction is harmless because SD rules are invariant to strictly positive affine transformation i.e.

)()( UbuUu for all 0>b 8 We focus on a definition in terms of utility functions, because we analyze the role of preference

assumptions. SD may be defined equivalently in terms of distribution functions or their quantiles (see

e.g. Levy, 1992, 1998). In fact, the LP dual of test statistic (9) formulates in terms of quantiles;

associated with each primal variablet , t , is a dual restriction on the running mean t

t

s

s /1

=

x .


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motivation for assuming market efficiency lies in the popularity of passive mutual

funds and exchange traded funds that track broad value-weighted equity indexes. In

other words, (some) investors reveal a preference for market indexes, and our

objective is to rationalize their choice and to analyze their preferences.9 Of course, we

could directly analyze the efficiency of actual funds. Still, for the sake of data

availability and comparability, we focus on the Fama and French market portfolio

(see Section III), which is used in many comparable studies (e.g. Harvey and Siddique

(2000) and Dittmar (2002)). Further, many actual funds, including total market index

funds based on the very broad Wilshire 5000 index (e.g. the Vanguard Total Stock

Market Index Fund) are likely to be very highly correlated with the Fama and French

market portfolio.

Special Cases

In our empirical analysis we will use various different SD criteria based on different

sets of preference assumptions. Table 1 summarizes the criteria, the assumptions and

the associated restrictions on marginal utility, as represented by r .

(Insert Table 1 about here)

The traditional criterion of Second-order Stochastic Dominance (SSD) assumes risk

aversion for the entire domain of returns, or equivalently risk aversion for losses, risk

aversion for gains and loss aversion (marginal utility of losses exceeds marginal

utility for gains):

(2) { }421 ,, SSD .10

9If the investment universe includes a riskless asset, then a sufficient condition for Assumption 3 is

that some investor holds a combination of the market portfolio and the riskless asset. Let

F )1( += for some ]1,0 , with F to denote a coordinate vector of zeros with a unity

value for the riskless asset F . If is the optimal solution for )( Uu , then is optimal relative

to )())1(()( + Uxxuxv F and hence efficient.10 Gains and losses are typically measured relative to a subjective reference point. For simplicity, we

set the reference point at zero. The use of excess returns implies that the reference point that

distinguishes gains from losses effectively equals the riskless rate. However, a follow-up analysis

demonstrates that the empirical results are not significantly affected by using a lower reference point ofzero or a higher reference point the average return on the market portfolio. If the reference point is not

known, it can be included as an additional model variable (at the cost of a possible loss of power).


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Models of decision-making and equilibrium under uncertainty traditionally use utility

functions from )( SSDU . However, as discussed in the Introduction, there is

compelling evidence that many decision-makers are risk seeking over a range.The psychological experiments by Kahneman and Tversky (1979) and

Kahneman and Tversky (1992) suggest that preferences are best described by a S-

shaped function that is convex for losses and concave for gains, and that is steeper for

losses than for gains (loss aversion). In recent years, Prospect Theory has attracted

much attention as a framework for understanding investor behavior and for explaining

financial market anomalies (see e.g. Benartzi and Thaler (1995), Barberis et al.

(2001), and Barberis and Huang (2001)). In this study, we consider two SD criteria

that are based on Prospect Theory. 11 Prospect Stochastic Dominance with Loss

Aversion (PSDL) assumes a S-shaped utility function with risk seeking for losses, risk

aversion for gains and loss aversion:

(3) { }431 ,, PSDL .

If we drop loss aversion, then we obtain Prospect Stochastic Dominance (PSD; Levy,

1998):

(4) { }43, PSD .

Recent experiments by Levy and Levy (2002) suggest that the Kahneman and

Tversky experiments may be biased by the research design. Specifically, bias may

originate from framing effects (unlike actual investments, the prospects involve either

only positive or only negative outcomes, but no mixed outcomes) and subjective

11 In contrast to Expected Utility Theory, Prospect Theory uses value functions with subjective decision

weights that overweight or underweight the true probabilities. Levy and Wiener (1998) demonstrate

that the PSD efficiency criterion is not affected by subjective transformations of the CDF that are

increasing and convex over losses and increasing and concave over gains, and hence PSD allows for

subjective underweighing of small probabilities of large gains and losses and overweighing of small

and intermediate probabilities of small and intermediate gains and losses. However, this pattern is

counterfactual (see e.g. Footnote 3) and rejection of PSD efficiency may mean rejection of S-shaped

probability transformations rather than S-shaped preferences. Still, there are good reasons to expect that

subjective probability distortion is less severe for investment choices than for some other choices. For

example, investors can extract information about the return distribution from historical return data and


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probability distortion (the prospects involve extremely small and large probabilities).

After correcting for these sources of bias, Levy and Levy find evidence against

Kahneman and Tversky type preferences; a large majority of subjects in their

experiments select PSD inefficient prospects. In fact, the Levy and Levy results

support a reverse S-shaped utility function with risk aversion for losses and risk

seeking for gains, i.e. exactly the opposite of Kahneman and Tversky type

preferences. Interestingly, Markowitz (1952) already suggested this type of utility

function. 12 In this study, we consider two SD criteria that build on Markowitz type

preferences.Markowitz Stochastic Dominance with Loss Aversion (MSDL) assumes a

reverse S-shaped utility function with risk aversion for losses, risk seeking for gains

and loss aversion:

(5) { }521 ,, MSDL .

If we drop loss aversion, then we obtain Markowitz Stochastic Dominance (MSD;

Levy and Levy, 2002):

(6) { }52 , MSD .

Interestingly, the classes )( MSDLU and )( MSDU include as special cases utility

functions that are consistent with Lopes (1987) SP/A Theory (introduced in portfolio

theory by Shefrin and Statman, 2000), where decision-makers are driven by the twin

desires for security and potential.

Figure 1 shows examples of non-concave utility functions that satisfy the

assumptions of PSDL, PSD, MSDL and MSD.

(Insert Figure 1 about here)

from fundamental economic data. In addition, if large amounts of money are at stake, then there is a

large incentive to gather and process such data so as to eliminate subjective probability distortions.12

Markowitz (1952) actually proposed a utility function with two Sshaped segments. Such a utility

function has a convex segment for large losses and a convex segment for small and moderate gains.

Hence, MSD is consistent with Markowitz type utility only if all outcomes are small or moderate

gains or losses. This assumption seems reasonable for our application, because we compare well-

diversified benchmark portfolios; for the full sample period (July 1963 October 2001), the minimummonthly excess return is -34.32 percent and the maximum excess return is 38.82 percent (see Table

2A-B).


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II. EMPIRICAL TESTING

In practical applications, the CDF generally is not known. Rather, information

typically is limited to a discrete set of time series observations, say )( 1 Txx L

with )( 1 Nttt xx Lx , and indexed by { }T

tt 1= .

Assumption 4 The observations are independent random draws from the CDF. 13

Since, by assumption, the timing of the draws is inconsequential, we are free to label

the observations by their ranking with respect to the evaluated portfolio, i.e.


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Linear Programming Formulation

We will test for efficiency by using the following test statistic:

(9) ),(

+

iTxitt

t

t 0/)(:min),(

x ,

with

(10) [{ }Rrst rststT

,,1),(:,:,1)( L= xx .

We can derive the following result for this test statistic:

THEOREM 1 Portfolio is empirically )(U -SD efficient if and only if

0),( =

The test statistic ),( basically checks if there exists a vector )( 1 T L that

represents the gradient vector at

X for a utility function )(Uu that satisfies

the first-order condition for portfolio optimality (see the proof in the Appendix). 16

The test statistic involves a linear objective function and a finite set of linear

constraints. Hence, the test statistic can be solved using straightforward Linear

Programming (LP). The model always has a feasible solution (e.g. e= ) and the

solution is bounded from below by zero (the case of efficiency) and bounded from

above by Tt

titi

/)(max

txx (the case with e= ). For small data sets up to

hundreds of observations and/or assets, the problem can be solved with minimal

computational burden, even with desktop PCs and standard solver software (like LP

solvers included in spreadsheets).Still, the computational complexity, as measured by

the required number of arithmetic operations, and hence the run time and memory

space requirement, increases progressively with the number of variables and

16

The theorem extends Posts (2001) Theorem 2 for SSD, and it exploits the result that the necessaryand sufficient first-order optimality condition applies not only for concave utility functions, but also formore general pseudoconcave utility functions (see e.g. Mangasarian, 1969).


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restrictions. Therefore, specialized hardware and solver software is recommended for

large-scale problems involving thousands of observations and/or assets.

The polyhedron )( involves constraints on pairs of observations ts, .

This suggests that the number of constraints increases quadratically with the numberof observations. However, many of the constraints are redundant, and the effective

number of constraints increases linearly with the number of observations. For

example, using z for the first observation in the domain of gains, i.e.

< zz xx 01 , it is easily verified that:

(11) { }1:)( 1 = TT

SSD L ;

(12) { }1;:)( 11 = TzzTT

PSDL LL ;

(13) { }1;1:)( 11 = TzzT

PSD LL ;

(14) { }TzTzT

MSD = LL 1;:)( 11 ;

(15) { }TzzT

MSDL = LL 1:)( 11 .

Statistical Inference

We have thus far discussed SD efficiency relative to the EDF )(xF rather than the

CDF )(xG . Generally, the EDF is very sensitive to sampling variation and the test

results are likely to be affected by sampling error in a non-trivial way. The applied

researcher must therefore have knowledge of the sampling distribution in order to

make inferences about the true efficiency classification. The remainder of this section

therefore develops an asymptotic hypothesis testing procedure for the test statistic

),( . The test procedure is based on two simplifications: (1) the use of a restrictive

null hypothesis and (2) the use of the limiting least favorable distribution. Our null

( 0H ) is that the assets have an equal and finite mean, i.e. ex =][E ,


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null is restrictive, because it gives a sufficient but not necessary condition for

efficiency. In fact, under the null, all portfolios are efficient, because they

optimize expected utility for xxu =)( , i.e. for risk neutral investors. The shape of the

distribution of )( under the null generally depends on the shape of )(xG . Ourapproach will be to focus on the least favorable distribution, i.e. the distribution that

maximizes the size or relative frequency of Type I error (rejecting the null when it is

true). This approach stems from the desire to be protected against Type I error. For

each )(xG , the size is always smaller than the size for the least favorable distribution.

Naturally, this approach comes at the cost of a high frequency of Type II error

(accepting the null when it is not true) or a low power (1- the relative frequency of

Type II error).

Using )( eC , we can summarize the asymptotic sampling distribution

of our SSD test statistic as follows: 17,18

THEOREM 2 For the asymptotic least favorable distribution, the p-value

]),(Pr[ 0Hy> , 0y , equals the integral ))(1(

ez

zy

d

, with )(z

for a N-

dimensional multivariate normal distribution function with zero means and

covariance matrix T/)( CC .

The theorem shows the crucial role of the length of the time series (T) and the length

of the cross-section (N); the p-values decrease as the time series grows, and increase

as the cross section grows. We may test efficiency by comparing the p-value for the

observed value of ),( with a predefined level of significance; we may reject

efficiency if thep-value is smaller than or equal to the significance level.Computing the p-value requires the unknown covariance terms ij . We may

estimate these parameters in a distribution-free and consistent manner using the

sample equivalents:

17We use for an identity matrix with dimensions conforming to the rules of matrix algebra.


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(16)

t t

jtjt

t

ititij TTxxTxx /)/)(/( .

We stress that the asymptotic p-values rely on a restrictive null and on the least

favorable distribution. For this reason, the p-values may involve low power in small

samples. An alternative approach is to approximate the sampling distribution by

means of bootstrapping. Bootstrapping can deal directly with the true null (SD

efficiency) and with the true distribution, and hence it can yield more powerful

results. Of course, this benefit has to be balanced against the additional computational

burden of using computer simulations.

III.ANALYZING AGGREGATE INVESTOR PREFERENCES

We analyze investor preferences by testing if different SD criteria (characterized by

different sets ) can rationalize the market portfolio. For this purpose, we need a

proxy for the market portfolio and proxies for the individual assets in the investment

universe. We will use the Fama and French market portfolio, which is the value-

weighted average of all NYSE, AMEX, and NASDAQ stocks. Further, we use the

one-month US Treasury bill as the riskless asset. Finally, for the individual risky

assets, we use three sets of benchmark portfolios:

1. The 25 Fama and French benchmark portfolios constructed as the intersections of

5 quintile portfolios formed on size and 5 quintile portfolios formed on BE/ME.

We use data on monthly returns (month-end to month-end) from July 1963 to

October 2001 (460 months) obtained from the data library on the homepage of

Kenneth French.19,20

Also, we analyze the stability of the results by applying thetest to 2 non-overlapping subsamples of equal length (230 months): (1) July

1963-August 1982 and (2) September 1982October 2001.

18 The theorem extends Posts (2001) Theorem 3 for SSD, and it exploits the result that ),( is

bounded from above by

Tx

t

titi

/)(max)( x (see the proof in the Appendix). This argument

applies not only forSSD , but also more generally for all .

19The data library is found at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french.

20

The data set starts in 1963 because the COMPUSTAT data used to construct the benchmarkportfolios are biased towards big historically successful firms for the earlier years (see Fama and

French, 1992).


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2. A set of 30 industry momentum portfolios constructed from 30 Fama and French

benchmark portfolios based on industry classification. The momentum portfolios

are based on the ranking of the returns over the past 6 months; momentum

portfolio no. 1 equals the Fama and French industry portfolio with the lowest past

return, and portfolio no. 30 equals the industry portfolio with the highest past

return. 21 The portfolios are rebalanced yearly in July. 22 As for the set of 25 size-

value portfolios, we use data on monthly excess returns (month-end to month-

end) from July 1963 to October 2001 (460 months), and we also consider the

subsamples July 1963-August 1982 and September 1982October 2001. The raw

data on the industry portfolios are obtained from the homepage of Kenneth

French.

3. The set of 27 benchmark portfolios described in Carhart et al. (1994) and used in

Carhart (1997). The portfolios are formed using a three-way classification based

on size, BE/ME and momentum. These portfolios are formed by dividing all

stocks into thirds based on B/M. Each of the three first-level portfolios is then

divided into three portfolios based on size. Finally, each of the nine second-level

portfolios is divided into past losers, middle and past winners. We use data

on monthly returns (month-end to month-end) from July 1963 to December 1994

(378 months), as well as the subsamples from July 1963 to March 1979 (189

months) and from April 1979 to December 1994 (189 months).

These three sets of benchmark portfolios provide a challenge, because many studies

indicate that the cross-sectional pattern of returns across size, BE/ME and momentum

portfolios cannot be explained by the traditional approach based on concave utility

functions. Tables 2A-C show some descriptive statistics for the excess returns of the

benchmark portfolios for the full sample.

(Insert Table 2A-C about here)

21 Similar results are obtained for portfolios based on the returns over the past 3, 9 and 12 months.


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16

To give some feeling for data, Figures 2A-C show mean-variance diagrams including

the individual benchmark portfolios (the bright dots), the market portfolio (M) and

the mean-variance efficient frontier (the line segment OA).23 Clearly, the market

portfolio is highly inefficient in terms of mean-variance analysis in all cases. Roughly

speaking, there exist portfolios with the same standard deviation as the market

portfolio but twice the average excess return, and there exist portfolios with the same

average excess return as the market portfolio but only half the standard deviation.

This result violates the central prediction of the MV CAPM: mean-variance efficiency

of the market portfolio.

(Insert Figure 2A-C about here)

The maintained preference assumptions of MV CAPM are a possible explanation for

this violation; variance does not fully capture the risk profile of assets unless investor

utility is quadratic. This provides the motivation for testing if the market portfolio is

efficient in terms of SD criteria. We employ the SD efficiency criteria discussed in

Section II: SSD, PSDL, PSD, MSDL and MSD. For each criterion, we compute the

value of the test statistic ),( and the associated asymptotic least favorable p-

value. We reject efficiency if the p-value is smaller then or equal to the significance

level of 10 percent. Tables 3A-C show the test results for the 3 sets of benchmark

portfolios. The results vary across the different sets of benchmark portfolios and the

different samples and subsamples. However, the relative goodness of the different SD

criteria is remarkably robust across benchmark portfolios and subsamples; PSDL and

PSD perform worst and MSD performs best.

The market portfolio is significantly SSD inefficient relative to the Fama and

French size-BE/ME portfolios and the Carhart size-BE/ME-momentum portfolios.

Based on this finding, we reject the SSD criterion. Harvey and Siddique (2000) and

Dittmar (2002) find that concave third-order and fourth-order polynomial utility

functions substantially better explain the cross-sectional variation of stock returns

22The literature generally analyzes momentum in firm-specific returns. However, Moskowitz and

Grinblatt (1999) demonstrate that momentum also exists in industry portfolios and they suggest that

industry momentum in fact accounts for much of firm-specific momentum.23

Note that this is the frontier for the case without short selling. Again, we focus on the case where

portfolio possibilities are described by all convex combinations of the individual assets (see Section I).Further, the frontier of risky assets (AB) tightly envelops the individual benchmark portfolios. This

reflects the high correlation between the Fama and French benchmark portfolios.


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17

than quadratic utility functions do. Our results suggest that no concave utility function

can rationalize the market portfolio. Under our maintained assumptions, this implies

that investors that hold the market portfolio are not globally risk averse and utility is

not everywhere concave, and we have to account for (local) risk seeking behavior.

This result is in line with the experimental results by Levy and Levy (2001); they find

that a majority of subjects are not globally risk averse, even when controlling for

effects of framing and probability distortion.

(Insert Table 3A-C about here)

Prospect Theory offers an alternative to the traditional approach based on concave

utility. However, we find strong evidence against two key elements of Prospect

Theory: risk seeking for losses and loss aversion. Both criteria that impose risk

seeking for losses (PSD and PSDL) are rejected all sets of benchmark portfolios. This

implies that no S-shaped utility function can rationalize the market portfolio. This

finding is in line with the experimental evidence by Levy and Levy (2002) that a large

majority of subjects select PSD inefficient prospects. Similarly, we reject all three

criteria that assume loss aversion (SSD, PSDL and MSDL). Barberis and Huang(2001) found that a model with loss aversion couldnt reproduce the empirical cross-

sectional pattern of stock returns. They used a simple utility function formed from one

linear line segment for losses and one for gains, and hence investors are assumed

locally risk neutral over gains and over losses. Our results suggest that their

conclusion is robust for more general preference assumptions; cross-sectional stock

returns are not consistent with loss aversion.

Only the MSD criterion is consistent with the data for all sets of benchmark

portfolios and for all samples and subsamples. This suggests that Markowitz type

reverse S-shaped utility functions can rationalize the market portfolio, and that risk

aversion over losses and risk seeking over gains helps to explain the cross-sectional

pattern of stock returns. If investors are risk averse for losses and risk seeking for

gains, then they will pay (ask) a premium for stocks that have low (high) downside

risk in bear markets and high (low) upside potential in bull market. Hence, stocks with

low (high) downside risk bear markets and high (low) upside potential provide low

(high) expected returns. This explanation is consistent with the experimental evidence


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18

by Levy and Levy (2002) that a large majority of subjects select MSD efficient

prospects. Our results suggest that actual stock returns are also consistent with this

explanation.

Further results

Further support for Markowitz type preferences can be found by analyzing the

relationship between average return and measures of downside risk and upside

potential. Two simple measures are downside beta (market beta for periods with a

negative market return) and upside beta (market beta for periods with a positive

market return); see e.g. Bawa and Lindenberg (1977). Risk aversion for losses

suggests a positive relation between expected return and downside beta, and risk

seeking for gains suggests a negative relation between expected return and upside

beta.For the sake of illustration, we estimated the downside and upside betas of the

27 Carhart benchmark portfolios. (Similar results are obtained for the 25 Fama and

French benchmark portfolios and the 30 industry momentum portfolios). Figure 3

shows the results.

(Insert Figure 3 about here)

Obviously, beta risk generally is highly asymmetric across falling and rising

markets.24 In addition, the differences between downside and upside betas seem to

support Markowitz type preferences, i.e. high downside beta portfolios generally have

high average returns, while high upside betas portfolios have low average returns. For

example, the high yield portfolio of small value winner stocks (with an average

monthly excess return of 1.341) has a downside beta of 1.409 and an upside beta of

0.883, while the low yield portfolio of the big growth loser stocks (with average

0.105) has a downside beta of 0.849 and an upside beta of 1.119. We may use

regression analysis to analyze if this pattern applies for the entire sample. Specifically,

using OLS regression analysis, we find the following cross-sectional relation between

24Several authors have found similar differences in upside and downside risk measures (see e.g. Ang

et al., 2001). These findings are sometimes interpreted as evidence for Prospect Theory. By contrast,we find evidence against two key elements of Prospect Theory -risk seeking for losses and loss

aversion- and our explanation rests on risk seeking for gains.


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19

average excess return

t

iti Tx / and estimated downside betai

and upside beta

+i

(standard deviations within brackets):25

(17) 601.0

)154.0(938.1

)060.0(856.0

)233.0(633.1

2 =+= + Riii .

The signs of the coefficients are consistent with Markowitz type preferences: a

positive coefficient for downside beta and hence risk aversion for losses, and a

negative coefficient for upside beta and hence risk seeking for gains. Of course, these

results may be sensitive to the parametric specification (a linear relationship between

expected return and the downside and upside betas) and to the estimation method

(estimation based on e.g. generalized methods of moments or maximum likelihood

procedures may give different results). Also, strictly speaking, we have to reject the

two-beta specification, because it does not give a perfect fit (the betas explain only 60

percent of the variation in average return) and because the intercept (the average

excess return of a zero-beta portfolio) is significantly higher than zero. Still, the

regression results do provide indirect support for our explanation based on risk

seeking for gains.

IV. CONCLUSIONS

We cannot reject MSD efficiency of the market portfolio relative to benchmark

portfolios formed on size, BE/ME and momentum. Hence, Markowitz type reverse S-

shaped utility functions may rationalize the market portfolio. By contrast, the

alternative SD criteria of SSD, PSDL, PSD and MSDL are rejected and utility

functions with global risk aversion, risk seeking over losses or loss aversion cannot

rationalize the market portfolio. These results suggest that the individual choice

behavior observed in the experiments by Levy and Levy (2002) can also help to

explain aggregate investor behavior. If investors are risk averse for losses and risk

seeking for gains, then they are willing to pay a premium for stocks that give

downside protection in bear markets and upside potential in bull markets. Our results

25

The cross-sectional regression gives a Fama and McBeth (1973) type test with betas assumedconstant over time. The assumption of constant betas is needed for comparison with the SD tests that

build on the assumption that the excess returns are serially IID (Assumption 4).


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20

suggest that this explanation is consistent with the cross-sectional pattern of stock

returns. Roughly speaking, stock returns suggests that investors are driven by the twin

desires for security and potential and investment portfolios are designed to avoid

poverty and to give a chance at riches (as in Shefrin and Statman, 2000).

Of course, our results may be biased by our maintained assumptions: we used

a single-period, portfolio-oriented model with a large number of expected utility

maximizers and without market frictions (apart from short selling restrictions). In

addition, we assumed a simple data generating process with a serially independent

and identical distribution for the excess returns. There are good reasons to doubt these

maintained assumptions. Further research could focus on relaxing our economic

assumptions (e.g. by considering the multiperiod consumption-investment problem,

imperfect competition, market imperfections like transaction costs and taxes, and non-

expected utility theories with bounded rationality and/or imperfect information) and

on relaxing our statistical assumptions (e.g. by using econometric time series

estimation techniques to estimate a conditional CDF). Still, at the very least, our

results suggest that Markowitz type utility functions are capable of capturing the

cross-sectional pattern of stock returns even under very simple economic and

statistical assumptions.

We hope that our results provide a stimulus for further research based on

Markowitz type of utility functions (and non-concave utility functions in general).

Also, we hope that this study contributes to the further proliferation of the SD

methodology. Since large, statistically representative samples often are available in

financial economics, the nonparametric SD approach is a useful complement to

existing parametric approaches.


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21

APPENDIX

Proof to Theorem 1: The problem

t

tTu /)(max

x , )(Uu , maximizes a

strictly increasing and hence pseudoconcave objective over a polytope. Hence,

is the optimal portfolio i.e.

t

t Tu /)(maxarg

x if and only if all assets are

enveloped by the tangent hyperplane, i.e.

(18)

iTxut

ittt0/))(( xx .

By construction, ))()(()( 1 Tuu xxuX L , )(Uu , is a feasible solution,

i.e. )()( Bu X . Optimality condition (18) implies that this solution is

associated with a solution value of zero. Hence, we find the necessary condition; is

)(U -SD efficient only if 0),( = .

To establish the sufficient condition, use )( 1* *

T

* L for the optimal

solution. It is easy to verify that we can always find some )(Uu with

*)( = Xu . For example, consider the piecewise-linear function

+


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22

Proof of Theorem 2: Since the unity vector is a feasible solution to the primal

problem, i.e. e , we know that ),(

Tx

t

titi

/)(max)( x . Known

results can derive the exact asymptotic sampling distribution of )( . Under the null

ex =][:0 EH , the tCx = ))()(( 1 tNttt xx xx L , t , are serially IID

random vectors with zero mean and covariance matrix CC . Hence, the

Lindeberg-Levy central limit theorem implies that the vector

T/eC

= )/)(/)(( 1 TxTxt

tNt

t

tt xx L obeys an asymptotically joint

normal distribution with zero mean and covariance matrix T/)( CC . Hence,

)( asymptotically behaves as the largest order statistic ofNrandom variables with

a multivariate normal distribution, and ])(Pr[ 0Hy> = [ ]ee yT /Pr1 C

asymptotically equals the multivariate normal integral

ez

zy

d )(1

. Since

),( )( , ]),(Pr[ 0Hy> is bounded from above by ])(Pr[ 0Hy> for

all return distributions )(xG . Moreover, it is easily verified that there exist )(xG for

which ),( approximates )( , and therefore the asymptotic distribution of )(

also represents the asymptotic least favorable distribution for ),( . Q.E.D.


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Barberis, N. M. Huang and T. Santos (2001): 'Prospect theory and asset prices',Quartely Journal of Economics 116, 1-53.

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Benartzi, S. and R. Thaler (1995): 'Myopic loss aversion and the equity premium

puzzle', Quartely Journal of Economics 110, 73-92.

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Carhart, M. M. (1997), On Persistence in Mutual Fund Performance, Journal of

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Carhart, M. M., R. J. Krail, R. I. Stevens, and K. E. Welch (1994), Testing the

conditional CAPM, Unpublished manuscript, University of Chicago.

Chapman, D. A. (1997): Approximating the Asset Pricing Kernel, Journal of Finance52 (4), 1383-1410.

Dittmar, R. (2002), Nonlinear Pricing Kernels, Kurtosis Preference, and Evidencefrom the Cross-Section of Equity Returns,Journal of Finance 57 (1), 369-403.

Fama and McBeth (1973): Risk, Return and Equilibrium : Empirical Tests, Journal ofPolitical Economy 71, 607-636.

Fama, E. and K. French (1992): The cross section of expected stock returns, Journal

of Finance 47 (2), 427-465.

Friedman, M., and Savage, L. (1948): The utility analysis of choices involving risk,Journal of Political Economy 56:279-304.

Friend, I. and Westerfield, R. (1980): Co-skewness and Capital Asset Pricing, Journal

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Ghysels, E. (1998), On Stable Factor Structures in the Pricing of Risk: Do Time-

Varying Betas Help or Hurt?,Journal of Finance 53 (2), 549-573.


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Hartley, R. and L. Farrell (2001): Can Friedman-Savage Utility Functions ExplainGambling?, forthcoming inAmerican Economic Review.

Harvey, C. R. and A. Siddique (2000): Conditional Skewness in Asset Pricing Tests,Journal of Finance 55 (3), 1263-1295.

Jagadeesh, N., and S. Titman (1993): Returns to buying winners and selling losers:Implications for stock market efficiency,Journal of Finance 48, 65-91.

Kahneman, D. and A. Tversky (1979): Prospect Theory: An analysis of decision

under risk,Econometrica 47:263-291.

Levy, H. (1969), Comment: A Utility Function Depending on the First ThreeMoments, Journal of Finance 24 (4), 715-719.

Levy, H. (1992): Stochastic Dominance and Expected Utility: Survey and Analysis,

Management Science 38(4), 555-593.

Levy, H. (1998): Stochastic Dominance, Kluwer Academic Publishers, Norwell, MA.

Levy, M. and H. Levy (2001), Testing for Risk Aversion: A Stochastic DominanceApproach,Economics Letters 71 (2), 233-240.

Levy, H. and Levy, M. (2002): Prospect Theory: Much Ado About Nothing?,forthcoming inManagement Science.

Levy, H. and Z. Wiener (1998), Stochastic Dominance and Prospect Dominance with

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Mangasarian, O. L. (1969):Nonlinear Programming, McGraw-Hill, New York.

Markowitz, H. (1952): The Utility of Wealth, Journal of Political Economy 60, 151-

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Sharpe, W. (1964): Capital asset prices: a theory of market equilibrium underconditions of risk,Journal of Finance 19, 425-442.

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of Finance 64, 489-509.

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Table 1: Preference Assumptions

The preference assumptions underlying the criteria of Second-order Stochastic Dominance (SSD),

Prospect Stochastic Dominance with Loss Aversion (PSDL), Prospect Stochastic Dominance (PSD),

Markowitz Stochastic Dominance with Loss Aversion (MSDL), and Markowitz Stochastic Dominance

(MSD). Each assumption, 5,,1L=r , is represented by a polyhedron 2r ; each assumption

effectively imposes the restriction )()( yuxu for all ryx ),( .

Criterion

r Assumptionr SSD PSDL PSD MSDL MSD

1 Loss Aversion+ X X X

2

Risk Aversion

for Losses yxyx :),(2 X X X

3

Risk Seeking

for Losses { }yxyx :),(2 X X

4

Risk Aversion

for Gains { }yxyx + :),(2 X X X

5

Risk Seeking

for Gains { }yxyx + :),(2 X X


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Figure 1: Example non-concave utility functions that are consistent with the assumptions of Prospect

Stochastic Dominance with Loss Aversion (PSDL), Prospect Stochastic Dominance (PSD), MarkowitzStochastic Dominance with Loss Aversion (MSDL), and Markowitz Stochastic Dominance (MSD).

MSDL


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Table 2A: Descriptive Statistics Size-BE/ME Portfolios

Monthly excess returns (month-end to month-end) from July 1963 to October 2001 (460 months) for

the value-weighted Fama and French market portfolio and 25 value-weighted benchmark portfolios

based on market capitalization (size) and/or book-to-market equity ratio (BE/ME). Excess returns are

computed from the raw return observations by subtracting the return on the one-month US Treasury

bill. All data are obtained from the data library on the homepage of Kenneth French.

Mean St. Dev. Skewness Kurtosis Minimum Maximum

Market Portfolio 0.462 4.461 -0.498 2.176 -23.09 16.05

Benchmark Portfolios

BE/ME Size

Growth Small 0.235 8.246 0.003 2.466 -34.32 38.82

2 Small 0.733 7.064 0.037 3.338 -31.30 36.67

3 Small 0.815 6.140 -0.087 3.233 -29.17 27.86

4 Small 0.998 5.724 -0.149 3.636 -29.47 27.43

Value Small 1.075 5.943 -0.098 4.128 -29.45 31.83

Growth 2 0.359 7.494 -0.316 1.673 -33.21 29.62

2 2 0.622 6.120 -0.472 2.778 -32.50 26.25

3 2 0.842 5.410 -0.488 3.739 -28.10 26.78

4 2 0.913 5.154 -0.385 3.979 -27.09 26.70

Value 2 0.966 5.660 -0.243 4.245 -29.83 29.43

Growth 3 0.380 6.908 -0.342 1.365 -29.96 23.39

2 3 0.665 5.511 -0.624 3.051 -29.65 23.35

3 3 0.673 4.962 -0.621 2.984 -25.56 21.14

4 3 0.818 4.710 -0.327 3.029 -23.20 22.61

Value 3 0.953 5.297 -0.423 4.384 -27.79 27.30

Growth 4 0.506 6.146 -0.182 1.683 -26.02 25.01

2 4 0.433 5.218 -0.571 3.317 -29.62 20.46

3 4 0.651 4.855 -0.423 3.348 -26.13 22.87

4 4 0.820 4.622 0.035 2.394 -18.30 24.57

Value 4 0.880 5.347 -0.198 2.697 -25.36 26.20

Growth Big 0.442 4.841 -0.177 1.676 -22.15 21.70

2 Big 0.441 4.588 -0.334 1.897 -23.21 16.05

3 Big 0.493 4.344 -0.243 2.639 -22.47 18.22

4 Big 0.603 4.289 0.056 1.574 -15.35 18.85

Value Big 0.583 4.645 -0.146 0.944 -19.28 15.39


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Table 2B: Descriptive Statistics Industry Momentum Portfolios

Monthly excess returns (month-end to month-end) from July 1963 to October 2001 (460 months) for

the value-weighted Fama and French market portfolio and 30 value-weighted benchmark portfolios

formed on industry momentum. The portfolios are constructed based on the past performance ranking

of 30 Fama and French industry portfolios; momentum portfolio no. 1 equals the Fama and French

industry portfolio with the lowest return over the past 6 months, and portfolio no. 30 equals the

industry portfolio with the highest past return. The portfolios are rebalanced yearly in July. Excess

returns are computed from the raw return observations by subtracting the return on the one-month US

Treasury bill. Raw data on industry portfolios are obtained from the data library on the homepage of

Kenneth French.

Mean St. Dev. Skewness Kurtosis Minimum Maximum



1 0.700 6.510 0.297 1.922 -26.43 29.20

2 -0.080 5.986 -0.442 2.205 -31.05 19.96

3 0.153 6.288 -0.402 3.070 -32.10 22.82

4 0.560 5.721 -0.252 2.436 -31.59 18.48

5 0.257 5.563 -0.152 1.398 -21.56 19.97

6 0.337 5.500 0.159 0.782 -19.69 19.92

7 0.218 5.558 -0.030 1.210 -21.08 21.77

8 0.259 6.276 -0.443 2.021 -28.40 25.18

9 0.333 5.828 -0.258 1.281 -21.66 19.08

10 0.408 5.925 -0.011 1.630 -22.25 25.91

11 0.623 5.486 0.021 3.714 -27.67 28.10

12 0.442 5.836 -0.404 3.033 -32.69 19.17

13 0.318 5.662 -0.239 2.394 -28.83 24.48

14 0.358 5.634 -0.098 1.616 -24.17 19.77

15 0.570 5.714 0.033 0.824 -20.90 19.56

16 0.566 5.972 0.318 1.523 -18.74 29.07

17 0.679 5.950 0.118 3.179 -28.60 31.84

18 0.775 5.776 -0.341 2.465 -31.96 22.6719 0.169 5.523 -0.502 1.852 -27.93 16.53

20 0.475 5.819 -0.415 2.496 -29.59 23.24

21 0.850 5.576 -0.706 3.297 -32.14 17.01

22 0.809 6.460 -0.142 3.500 -33.22 30.36

23 0.491 5.676 -0.252 1.372 -26.40 18.57

24 0.485 5.586 -0.303 1.186 -25.91 17.14

25 0.657 5.529 -0.358 2.385 -28.60 21.84

26 0.680 6.077 0.096 1.901 -19.31 28.44

27 0.683 6.456 0.640 6.429 -28.70 45.92

28 0.986 6.152 -0.082 3.037 -33.02 28.19

29 1.092 7.030 -0.450 2.078 -31.76 23.56

30 0.949 7.319 0.178 2.823 -32.91 37.57


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Table 2C: Descriptive Statistics Size-Value-Momentum Portfolios

Monthly excess returns from July 1963 to December 1994 (378 months) for the value weighted Fama

and French market portfolio and 27 value weighted benchmark portfolios based on market

capitalization (size), book-to-market equity (BE/ME) and momentum. The 27 portfolios are formed by

dividing all stocks into thirds based on B/M. Each of the three first-level portfolios is then divided into

three portfolios based on size. Finally, each of the nine second-level portfolios is divided into past

losers, middle and past winners. Excess returns are computed from the raw return observations by

subtracting the return on the one-month US Treasury bill. Data on the 27 portfolios are courtesy of

Mark Carhart. The remaining data on the market portfolio and the Treasury bill. All other data are

obtained from the homepage of Kenneth French.

Mean St. Dev Skewness Kurtosis Minimum Maximum



BE/ME Size Momentum

Growth Small Loser -0.279 6.644 0.003 2.784 -31.41 30.45

Neutral Small Loser 0.306 5.975 0.474 5.363 -24.81 37.51

Value Small Loser 0.552 6.440 0.974 7.774 -28.26 45.34

Growth Small Middle 0.349 6.070 -0.461 3.194 -31.33 27.54Neutral Small Middle 0.631 4.957 -0.201 5.038 -27.05 27.02

Value Small Middle 1.062 5.583 0.199 6.219 -29.09 33.99

Growth Small Winner 0.933 6.755 -0.704 2.492 -33.52 19.45

Neutral Small Winner 1.157 6.088 -0.730 3.569 -32.60 23.17

Value Small Winner 1.341 6.246 -0.289 4.564 -32.00 30.20

Growth Medium Loser -0.089 5.952 0.088 2.757 -26.49 27.54

Neutral Medium Loser 0.400 5.396 0.501 3.468 -19.08 31.20

Value Medium Loser 0.582 6.078 0.469 4.678 -25.79 38.07

Growth Medium Middle 0.143 5.341 -0.461 2.207 -26.81 18.96

Neutral Medium Middle 0.513 4.498 -0.344 4.720 -25.68 23.05

Value Medium Middle 0.909 5.303 -0.016 6.042 -28.30 31.03

Growth Medium Winner 0.895 6.033 -0.550 2.261 -29.90 20.73Neutral Medium Winner 0.772 5.325 -0.896 3.683 -30.54 17.52

Value Medium Winner 1.287 5.881 -0.806 5.234 -33.86 25.76

Growth Big Loser 0.105 5.218 0.178 2.208 -20.40 25.40

Neutral Big Loser 0.438 4.801 0.477 2.287 -19.66 21.74

Value Big Loser 0.600 5.569 0.718 4.414 -16.99 35.34

Growth Big Middle 0.250 4.507 -0.161 1.885 -20.73 17.86

Neutral Big Middle 0.367 4.211 0.158 2.307 -15.57 21.03

Value Big Middle 0.589 4.752 -0.053 2.717 -23.47 20.41

Growth Big Winner 0.611 5.348 -0.298 2.035 -23.68 21.21

Neutral Big Winner 0.595 4.856 -0.347 2.565 -24.00 19.95

Value Big Winner 0.923 5.352 -0.254 2.846 -24.78 22.84


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Figure 2A: Mean-variance diagram for the historical excess returns of

the 25 Fama and French size -BE/ME portfolios (the bright dots) and the

Fama and French market portfolio (M). OA represents the efficient frontier

(with a the US Treasury bill, short sales not allowed).

M

A

O

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 2 4 6 8 10

standard deviation

meanexcessreturn


34/40

32

Figure 2B: Mean-variance diagram for the historical excess returns of

the 30 industry momentum portfolios (the bright dots) and the Fama and

French market portfolio (M). OA represents the efficient frontier (with a

the US Treasury bill, short sales not allowed).

A

M

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.01.2

1.4

0 2 4 6 8 10

standard deviation

meanexcessreturn

O


35/40

33

Figure 2C: Mean-variance diagram for the historical excess returns of

the 27 Carhart size-BE/ME-momentum portfolios (the bright dots) and the

Fama and French market portfolio (M). OA represents the efficient frontier

(with a the US Treasury bill, short sales not allowed).

M

-0.4

-0.2

0.0

0.2

0.4

0.6

0.81.0

1.2

1.4

0 2 4 6 8 10

standard deviation

meanexcessreturn

O


36/40

34

Table 3A: Test Results Size-Value Portfolios

We test whether the Fama and French market portfolio is )(U -SD efficient relative to

all portfolios formed from a US Treasury bill and 25 benchmark portfolios formed on

market capitalization (size) and book-to-market-equity ratio (BE/ME). Each cell shows

the observed value for the test statistic ))(,( , as well as the asymptotic leastfavorable p-value (within brackets). We use a bold font if results are statistically

significant at a level of significance of 90 percent.

Jul 1963 -

Oct 2001

Jul 1963 -

Aug 1982

Sep 1982 -

Oct 2001

SSD 0.434

(0.031)

0.585

(0.034)

0.299

(0.490)

PSDL 0.613

(0.002)

0.918

(0.000)

0.387

(0.243)

PSD 0.432

(0.031)

0.905

(0.000)

0.356

(0.324)

MSDL 0.434

(0.031)

0.585

(0.034)

0.299

(0.490)

MSD 0.207

(0.501)

0.226

(0.748)

0.192

(0.865)


37/40

35

Table 3B: Test Results Industry Momentum Portfolios



industry momentum. Each cell shows the observed value for the test statistic ))(,( ,

as well as the asymptotic least favorable p-value (within brackets). We use a bold font if

results are statistically significant at a level of significance of 90 percent.

Jul 1963 -

Oct 2001

Jul 1963 -

Aug 1982

Sep 1982 -

Oct 2001

SSD 0.376

(0.375)

0.604

(0.214)

0.492

(0.479)

PSDL 0.595

(0.026)

0.995

(0.011)

0.484

(0.508)

PSD 0.595

(0.026)

0.749

(0.057)

0.350

(0.915)

MSDL 0.376

(0.375)

0.604

(0.214)

0.492

(0.479)

MSD 0.344

(0.510)

0.568

(0.295)

0.356

(0.892)


38/40

36

Table 3C: Test Results Size-Value-Momentum Portfolios



size, value and momentum. Each cell shows the observed value for the test statistic

))(,( , as well as the asymptotic least favorablep-value (within brackets). We use a

bold font if results are statistically significant at a level of significance of 90 percent.

Jul 1963 -

Dec 1994

Jul 1963 -

Mar 1979

Apr 1979

Dec 1994

SSD 0.469

(0.013)

0.780

(0.004)

0.442

(0.169)

PSDL 0.954

(0.000)

1.171

(0.000)

0.737

(0.004)

PSD

0.861(0.000)

0.930

(0.000)

0.737

(0.004)

MSDL 0.469

(0.013)

0.780

(0.004)

0.442

(0.169)

MSD 0.291

(0.275)

0.346

(0.547)

0.384

(0.300)


39/40

37

Figure 3: Upside and downside market betas for the 27 Carhart size-BE/ME-momentum portfolios.

The dark dots represent the 9 portfolios with the highest average returns, the bright dots are the 9

portfolios with the lowest average returns and the grey dots are the remaining 9 portfolios with medium

average returns. Betas are computed using OLS regression and using the full samples of monthly

excess returns from July 1963 to December 1994.

0

0.5

1

1.5

2

0 0.5 1 1.5 2

upside beta

downsidebeta


40/40

Publications in the Report Series Research in Management

ERIM Research Program: Finance and Accounting

2002

A Stochastic Dominance Approach to SpanningThierry PostERS-2002-01-F&A

Testing for Third-Order Stochastic Dominance with Diversification PossibilitiesThierry PostERS-2002-02-F&A

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Modeling the Conditional Covariance between Stock and Bond Returns: A Multivariate GARCH ApproachPeter De Goeij & Wessel MarqueringERS-2002-11-F&A

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Does Risk Seeking Drive Asset Prices? A Stochastic Dominance Analysis of Aggregate Investor PreferencesThierry Post & Haim LevyERS-2002-50-F&A

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Reinventing The Hierarchy, The Case Of The Shell Chemicals Carve-OutMichel A. van den Bogaard, Roland F. SpeklERS-2002-52-F&A

*

A complete overview of the ERIM Report Series Research in Management:http://www.ers.erim.eur.nl

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